\(A\in \left(X,\left(\bar{N}_{p}^{q} \right)\right)\)  if and only if (REF ) holds and
\(\lim _{m\rightarrow \infty } \left({1\over R_m}\sum _{n=0}^{m}p_{m-n} q_nA_n \left(c^{(k)}\right)\right)=\alpha _k ~~~\forall ~k=0,1,2\ldots \) 
Hausdorff Measure of Noncompactness
Let \(S\)  and \(M\)  be the subsets of a metric space \((X,d)\)  and \(\epsilon >0\) . Then \(S\)  is called an \(\epsilon -\) net of \(M\)  in \(X\)  if for every \(x\in M\)  there exists \(s\in S\)  such that \(d(x,s)<\epsilon \) . Further, if the set \(S\)  is finite, then the \(\epsilon -\) net \(S\)  of \(M\)  is called finite \(\epsilon -\) net of \(M\) . A subset of a metric space is said to be totally bounded if it has a finite \(\epsilon -\) net for every \(\epsilon >0\)  [1]}.
If \(\mathcal {M}_X\)  denotes the collection of all bounded subsets of metric space \((X,d)\) . If \(Q\in \mathcal {M}_X\)  then the Hausdorff Measure of Noncompactness of the set \(Q\)  is defined by
\(\chi (Q)=\inf \left\lbrace \epsilon >0: Q \text{ has a finite } \epsilon -\text{net in }X \right\rbrace  \) 
The function \(\chi :\mathcal {M}_X\rightarrow [0,\infty )\)  is called Hausdorff Measure of Noncompactness [2]}
The basic properties of Hausdorff Measure of Noncompactness can be found in ([3]}, [4]}, [2]}).
Some of those properties are
If \(Q,Q_1\)  and \(Q_2\)  are bounded subsets of a metric space \((X,d),\)  then
\(\chi (Q)&=0 \Leftrightarrow Q~\text{ is totally bounded set,}\\\chi (Q)&=\chi (\bar{Q}),\\Q_1\subset Q_2 &\Rightarrow \chi (Q_1)\le \chi (Q_2),\\\chi (Q_1\cup Q_2)&=\max \left\lbrace  \chi (Q_1),\chi (Q_2)\right\rbrace ,\\\chi (Q_1\cap Q_2)&=\min \left\lbrace  \chi (Q_1),\chi (Q_2)\right\rbrace .\) 

Further if \(X\)  is a normed space then Hausdorff Measure of Noncompactness \(\chi \)  has the following additional properties connected with the linear structure.
\(\chi (Q_1+ Q_2)&\le \chi (Q_1)+\chi (Q_2)\\\chi (\eta Q)&=|\eta |\chi (Q)~~~~~~~~~~~~~~~~\eta \in \mathbb {C}\) 

The most effective way of characterizing operators between Banach Spaces is by applying Hausdorff Measure of Noncompactness. If \(X\)  and \(Y\)  are Banach spaces, and \(L\in \mathcal {B}(X,Y)\) , then the Hausdorff Measure of Noncompactness of \(L\) , denoted by \(\Vert L\Vert _\chi \)  is defined as
\(\Vert L\Vert _\chi =\chi \left(L(S_X)\right)\) 
Where \(S_X=\lbrace x\in X:\Vert x\Vert =1\rbrace \)  is the unit ball in \(X\) .
From ([6]}, Corollary 1.15) we know that
\( L ~\text{ is compact if and only if } ~\Vert L\Vert _\chi =0 \) 
.
Proposition 3  ([2]}, Theorem 6.1.1, \(X=c_0\) )
Let \(Q\in M_{c_0}\)  and \(P_r:c_0\rightarrow c_0 ~~(r\in \mathbb {N}\)  be the operator defined by \(P_r(x)=(x_0,x_1,\ldots , x_r, 0,0,\ldots )\)  for all \(x=(x_k)\in c_0\) . Then, we have
\(\chi (Q)=\lim _{r\rightarrow \infty }\left(\sup _{x\in Q}\Vert (I-P_r)(x)\Vert  \right)\) 
where \(I\)  is the identity operator on \(c_0\) .

Proposition 4  (cf. [2]}, Theorem 6.1.1)
Let \(X\)  be a Banach space with a Schauder basis \(\lbrace e_1,e_2,\ldots \rbrace \) , and \(Q\in M_{X}\)  and \(P_n:X\rightarrow X ~~(n\in \mathbb {N}\)  be the projector onto the linear span of \(\lbrace e_1,e_2,\ldots , e_n\rbrace \) . Then, we have
\({1\over a}\lim _{n\rightarrow \infty }\sup &\left(\sup _{x\in Q}\Vert (I-P_n)(x)\Vert  \right) \le \chi (Q)\\&\le \inf _{n}\left(\sup _{x\in Q}\Vert (I-P_n)(x)\Vert  \right)\le \lim _{n\rightarrow \infty }\sup \left(\sup _{x\in Q}\Vert (I-P_n)(x)\Vert  \right)\) 
where \(a=\lim _{n\rightarrow \infty }\sup \Vert I-P_n\Vert \) , and \(I\)  is the identity operator on \(c\) .
If \(X=c\)  then \(a=2\) . (see [2]}, p.22).


Compact operators on the spaces \(\left(\bar{N}_{p}^{q} \right)_0\) , \(\left(\bar{N}_{p}^{q} \right)\)  and \(\left(\bar{N}_{p}^{q} \right)_\infty \)
Theorem 5 
Consider the matrix \(A\)  as in Corollary REF , and for any integers n,s, \(n>s\)  set
\(\Vert A\Vert ^{(s)}=\sup _{n>s}\sup _{m}\left\lbrace \sum _{k=0}^{m} R_{k}\left|\sum _{j=k}^{m} \left(-1\right)^{j-k}{H_{j-k}^{(p)}\over q_{j}}a_{nj}\right|\right\rbrace \) 
If \(X\)  be either \(\left(\bar{N}_{p}^{q} \right)_0\)  or \(\left(\bar{N}_{p}^{q} \right)\)  and \(A\in (X,c_0) \) . Then
\(\Vert L_A\Vert _{\chi }=\lim _{s \rightarrow \infty } \Vert A\Vert ^{(s)}.\) 
If \(X\)  be either \(\left(\bar{N}_{p}^{q} \right)_0\)  or \(\left(\bar{N}_{p}^{q} \right)\)  and \(A\in (X,c) \) . Then
\({1\over 2}\cdot \lim _{s\rightarrow \infty } \Vert A\Vert ^{(s)}\le \Vert L_A\Vert _{\chi }\le \lim _{r\rightarrow \infty } \Vert A\Vert ^{(s)}.\) 
and if \(X\)  be either \(\left(\bar{N}_{p}^{q} \right)_0\)  , \(\left(\bar{N}_{p}^{q} \right)\)  or \(\left(\bar{N}_{p}^{q} \right)_\infty \)  and \(A\in (X,\ell _\infty ) \) . Then
\(0\le \Vert L_A\Vert _{\chi }\le \lim _{s\rightarrow \infty } \Vert A\Vert ^{(s)}.\) 

Let \(F=\lbrace x\in X:\Vert x\Vert \le 1\rbrace \)  if \(A\in (X,c_0) \)  and \(X\)  is one of the spaces \(\left(\bar{N}_{p}^{q} \right)_0\)  or \(\left(\bar{N}_{p}^{q} \right)\) , then by Proposition  REF
\(\Vert L_A\Vert _{\chi }=\chi (AF)=\lim _{s \rightarrow \infty } \left[\sup _{x\in F}\Vert (I-P_s)Ax\Vert \right]\) 
Again using Proposition REF  and Corollary REF  we have
\(\Vert A\Vert ^{s}=\sup _{x\in F}\Vert (I-P_s)Ax\Vert \) 
From (REF ) and (REF ) we get
\(\Vert L_A\Vert _{\chi }=\lim _{s \rightarrow \infty } \Vert A\Vert ^{(s)}.\) 

Since every sequence \(x=(x_k)_{k=0}^{\infty } \in c\)  has a unique representation
\(x=le+\sum _{k=0}^{\infty } (x_k-l)e^{(k)}~~~~~~~~~~~\text{where} ~~l\in \mathbb {C}~~\text{is such that } x-le\in c_0\) 
We define \(P_s:c\rightarrow c\)  by \(P_s(x)=le+\sum _{k=0}^{s} (x_k-l)e^{(k)}\) , \(s=0,1,2,\ldots \) .
Then \(\Vert I-P_s\Vert =2\)  and using (REF ) and Proposition REF  we get
\({1\over 2}\cdot \lim _{s\rightarrow \infty } \Vert A\Vert ^{(s)}\le \Vert L_A\Vert _{\chi }\le \lim _{s\rightarrow \infty } \Vert A\Vert ^{(s)}\) 

Finally we define \(P_s:\ell _\infty \rightarrow \ell _\infty \)  by \(P_s(x)=(x_0,x_1,\ldots , x_s, 0,0\ldots )\) , \(x=(x_k)\in \ell _\infty \) .
Clearly \( AF\subset P_s(AF)+(I-P_s)(AF)\)
So using the properties of \(\chi \)  we get
\(\chi (AF)&\le \chi [P_s(AF)]+\chi [(I-P_s)(AF)]\\&=\chi [(I-P_s)(AF)] \\&\le \sup _{x\in F}\Vert (I-P_s)A(x)\Vert \) 

Hence by Proposition REF  and Corollary REF  we get
\(0\le \Vert L_A\Vert _{\chi }\le \lim _{s\rightarrow \infty } \Vert A\Vert ^{(s)}\)
A direct corollary of the above theorem is
Corollary 3 Consider the matrix \(A\)  as in Corollary REF , and \(X=\left(\bar{N}_{p}^{q} \right)_0\)  or \(X=\left(\bar{N}_{p}^{q} \right)\) 
then if \(A\in (X,c_0)\)  or \(A\in (X,c)\)  we have
\(L_A \text{ is compact if and only if } \lim _{s\rightarrow \infty }\Vert A\Vert ^{(s)}=0\) 

Further, for \(X=\left(\bar{N}_{p}^{q} \right)_0\)  , \(X=\left(\bar{N}_{p}^{q} \right)\)  or \(X=\left(\bar{N}_{p}^{q} \right)_\infty \) , if \(A\in (X,\ell _\infty )\)  then we have
\(L_A \text{ is compact if } \lim _{s\rightarrow \infty }\Vert A\Vert ^{(s)}=0\) 

In (REF ) it is possible for \(L_A\)  to be compact although \(\lim _{s\rightarrow \infty }\Vert A\Vert ^{(s)}\ne 0\) , that is the condition is only sufficient condition for \(L_A\)  to be compact.
For example, let the matrix \(A\)  be defined as \(A_n=e^{(1)}~~~n=0,1,2,\ldots \)  and the positive sequences \(q_n=3^n~,~n=0,1,2,\ldots \)  and \(p_0=1, p_1=1, p_k=0~,~\forall ~k=2,3,\ldots \)  .
Then by (REF ) we have
\(\sup _{n,m}\left\lbrace \sum _{k=0}^{m} R_{k}\left|\sum _{j=k}^{m} \left(-1\right)^{j-k}{H_{j-k}^{(p)}\over q_{j}}a_{nj}\right|\right\rbrace =\sup _{m}\left( 2-{2\over 3^{m}}\right)=2<\infty \) 
Now by Corollary REF  we know \(A\in \left( \left(\bar{N}_{p}^{q} \right)_\infty ,\ell _\infty \right)\)  .
But
\(\Vert A\Vert ^{(s)}=\sup _{n>s}\left[ 2-{2\over 3^{m}} \right]=2-{1\over 2\cdot 3^{s}}~~~~~~\forall ~s\) 
Which gives \(\lim _{s\rightarrow \infty }\Vert A\Vert ^{(s)} =2\ne 0\) .
Since \(A(x)=x_1\)  for all \(x\in \left(\bar{N}_{p}^{q} \right)_\infty \) , so \(L_A\)  is a compact operator.
